LOW-DEGREE EQUATIONS: A RAPID SURVEY

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1 LOW-DEGREE EQUATIONS: A RAPID SURVEY PAUL HEWITT, UNIV OF TOLEDO Liner systems. In ncient Irq even efore Hmmuri, nd indeed cross the ncient world, people hd figured out how to solve system of two low-degree equtions in two unknowns. For exmple, they knew how to determine pir of numers given their sum nd difference. In symols: (1) + = S, = D. In fct, the solution is esy to visulize. The verge of nd is 1 2S, which lies hlfwy etween nd : D/2 D/2 Hence S/2 (2) = 1 2 S D, = 1 2 S 1 2 D. Qudrtic systems. Now we move one step up in complexity: determine the vlue of two numers given their product nd difference. In symols: (3) = P, = D Let s solve it in the time-honored fshion: reduce prolem to simpler one we ve lredy solved. We ll use the identity (4) ( + ) 2 = ( ) in order to first determine the sum S = + from D nd P. We cn then use (2). Here is picture of this identity: Dte: 7 Septemer

2 2 PAUL HEWITT, UNIV OF TOLEDO Crry out the steps outlined ove. From the identity (4) you will find tht 1 (5) 2 S = 2 1 D2 + 4P = ( 1 2 D)2 + P. Next, pply (2) nd (5) to the qudrtic system (3). You will otin n ncient version of the qudrtic formul: (6) = ( 12 D)2 + P + 12 D, = ( 1 2 D)2 + P 1 2 D. The pythgoren theorem. The identity (4) is closely relted to n ncient proof of Pythgors Theorem, which ws in fct known long efore Pythgors, not only in ncient Irq, ut lso in ncient Indi nd ncient Chin. If c is the digonl of the rectngle, then when we look t this ( + ) ( + ) squre in two wys: we see tht ( + ) 2 = c nd lso tht ( + ) 2 = Hence c 2 = A specil cse the cse = ppered in the Sulvsutrs, from the Vedic period of Indi: roughly 1500 to 500. In fct, the discussion of this theorem mkes references to rick-mking, which ws not technology of the Indo-Europen Aryn (= Irnin) peoples who dominted the Vedic er, ut ws importnt to the erlier Drvidin people of the Hrppn culture, which flourished millennium efore in the Indus vlley. The theorem ws clled the Kou-Ku Theorem ( leg-thigh theorem ) in the very ncient Nine Chpters on the Mthemticl Art (Jiuzhng sunchu). Noody knows exctly when this text ws first written, ut most of the mteril ppers to dte from the Zhou Dynsty: 1100 to 256. When Qin Shi Hungdi unified Chin in 221 he ordered most ooks urned. (Qin is uried in Xin with the fmous terr cott rmy.) The Qin Dynsty did not lst long, ut ws followed y the Hn Dynsty: 207 to This ws the ge of empires, with long-lived dynsties stretching from the Atlntic to the Pcific. Pythgors ws religious mystic who lived in Greek colony in Croton, t the southern tip of Itly, no erlier thn 580. Greek golden ge : 500 to 300. His followers found deep pprecition of the divine in the mnifesttion of numer in the nturl world. Pythgors if he ws more thn legend lerned mthemtics where other erly Greek scholrs did, in either the Levnt or Egypt.

3 LOW-DEGREE EQUATIONS 3 Diophntos method. If discussion of qudrtic equtions is to e useful teching device, it needs to include lots of exmples tht re within the scope of students clculting ility. Hence it is useful to e le to generte lots of pythgoren triples : integer or rtionl solutions,, c to the pythgoren eqution = c 2. In fct, if we let p = /c nd q = /c then we see tht we re looking for rtionl points (p, q) on the circle x 2 + y 2 = 1. A fmous cly tlet Plimpton 322, from ner Ur out 1700, gives long list of such pythgoren triples. We don t know exctly how they generted such list, ut much lter work of Diophntos ( 200? 100? +100? +200?) of Alexndri shows how it could e done using the lger nd geometry known since ncient time. (p,q) y = m(x + 1) ( 1,0) Diophntos proves tht there is one-to-one correspondence etween the rtionl points on this circle nd the lines of rtionl slope through the point ( 1, 0). The eqution of ny such line is y = m(x + 1). Hence the intersection of the line nd the circle is determined y the system of equtions { } x (7) 2 + y 2 = 1 y = m(x + 1) Sustitute the second eqution into the first nd you find tht (8) (1 + m 2 )x 2 + 2m 2 x + m 2 = 1. Bring ll of the terms to the left-hnd side nd divide y the leding coefficient. You will otin qudrtic polynomil which fctors s (x + 1)(x p), since 1 nd p re the x-coordintes of the two points of intersection. When you expnd the product nd compre the two expressions you will find tht (9) p 1 = 2m2 1 + m 2. Thus (10) p = 1 m2 2m, q = m(p + 1) = 1 + m2 1 + m 2. This is simply resttement of our qudrtic identity: (1 m 2 ) 2 + 4m 2 = (1 + m 2 ) 2.

4 4 PAUL HEWITT, UNIV OF TOLEDO Diophntos used this secnt method nd lso its nturl limit, the tngent method to mke n extensive study of rtionl solutions of indeterminte qudrtic nd cuic equtions. He pplied his secnt-nd-tngent method to cuic curve to produce new solutions from old. If we hve rtionl points P 1 nd P 2 on the cuic curve C then the line L = P 1 P 2 meets C in third rtionl point P 3. It turns out tht this defines kind of ddition on cuic: if we let O e the point t infinity, then it is the dditive identity for n elin group where P 1 + P 2 + P 3 = O whenever L is line nd C L = {P 1, P 2, P 3 }. Such curve with this group lw is clled n elliptic curve. Elliptic curves re centrl to wide rnge of modern mthemtics, from numer theory to differentil equtions. Diophntos work is so different thn tht of ll other Greek mthemticins tht it ppers to hve een ignored y ll of his contemporries. His work ws studied y the lter Aric scholrs of the Islmic Empire: 632 to It is from Aric trnsltions nd commentries tht we know Diophntos work. Indi, Bghdd, nd eyond. In Indi, indeterminte qudrtic equtions x 2 y 2 = were solved y y Brhmgupt (598 to 668) nd Bhskrchry (1114 to 1185), using methods they clled chkrvl. Their work influenced the scholrs t the House of Wisdom. These equtions re now clled Pell s equtions, nd the method is now prt of the theory of continued frctions. In Indi the method ws prt of trdition of pproximtions using lgeric identities which culminted in development of infinite series nd products, centuries efore their re-invention in Europe. At ny rte, Pell s eqution is so-nmed ecuse Fermt (1601 to 1665) invited Pell nd others to solve it. Fermt ws n mteur French mthemticin who ws inspired y recent trnsltion of Diophntos works into Ltin nd other Europen lnguges. Into the mrgin of this ook he is sid to hve written tht he hd found truly mrvelous proof tht when n > 2 there re no positive-integer solutions to the eqution n + n = c n, ut tht the mrgin ws too smll for it. Fermt worked t the eginning of one of the gretest revolutions in science nd mthemtics, leding to the development of clculus in the Age of Enlightenment, (1688 to 1789). Two key steps were the doption of the Indin deciml nottion for numers, nd the introduction of symolic lger. It took Europe long time to dopt deciml nottion. Fioncci nd others hd lerned it from the Ars, nd worked hrd to gin its cceptnce. During this period, from the lte Medievl Period to the Renissnce, round 1300 to 1600, more nd more mthemtics ws written in vernculr lnguges. (This my hve een in lrge prt due to the derth of Ltin tutors fter the Blck Plgue hd swept Europe strting in the 1340 s: million Europens died, t lest 30% of the popultion.) One of the things which intrigued the Europens, ws extending the qudrtic formul to equtions of higher degree. Pcioli (1445 to 1517) wrote pessimistic ccount of ll such ttempts, to tht point filures, in his Summ de rithmetic, geometri, proportioni et proportionlit, pulished But within few decdes oth the generl cuic nd qurtic equtions hd een solved! These results were pulished y Girolmo Crdno (1501 to 1576), nd hence they re known s Crdno s Formuls. He explined the cuic eqution using geometric resoning nlogous to the method of completing the squre, which ws known throughout the ncient world. More on this lter.

5 LOW-DEGREE EQUATIONS 5 Although Crdno himself recognized tht the formul ws useless in prctice, he noted curious consequence which would e the impetus to rethtking dvnces in mthemtics. He noticed tht even when the roots of n eqution re positive rel numers the formul involves, in its intermedite steps, the use of imginry squre-roots of negtives! In the end ll of the imginries cncel, ut why re they necessry in the formul? The success for cuics nd qurtics led to the hope tht ll polynomil equtions could e solved y rdicls. Using these newfngled complex numers Euler (1707 to 1783) nd Lgrnge (1736 to 1813) showed tht every polynomil fctors completely. But they could not find explicit formuls for the roots. Glois (1811 to 1832), uilding on work of Ael (1802 to 1829), found precise conditions under which polynomil eqution could e solved y rdicls. When n > 4 generlly it cnnot. The proof introduced the structure of group of permuttions, reflecting ll of the symmetries of the roots. At this point the history of lger turned from the lgorithmic to the strct. It is only now turning ck, ut not completely, with the dvent of computers. In the 19th century the new strct lger of groups nd rings nd fields nd idels produced stunning results. But none of them could crck the lowly Fermt s Lst Theorem. The new ide needed for this cme only in the 20th century, when Tniym (1927 to 1958) put forth series of conjectures, lter refined y Shimur (orn 1930) concerning the prmetriztion of elliptic curves. His conjectures were linked to Fermt s Lst Theorem y conjectures of Serre (orn 1926) concerning Glois representtions. The Tniym-Shimur conjectures were solved y Andrew Wiles (orn 1953), therey settling Fermt s Lst Theorem. Exercises. (1) Crry out the steps of the solution to the qudrtic system (3): use the identity (4) nd the solution (2) to first deduce eqution (5) nd then the qudrtic formul (6). (2) Explin in detil ll of the steps of the ncient Chinese proof of the Kou-Ku theorem (pythgoren theorem). (3) Crry out the steps in Diophntos method. Ech prt elow counts s seprte prolem, ut you must do ll three to get ny credit. () Verify tht the line through ( 1, 0) nd (p, q) stisfies the eqution y = m(x + 1). Deduce tht if p nd q re oth rtionl, then so is m. () Show tht the system (7) reduces to the single eqution (8). Put this eqution in stndrd form, y ringing ll the terms to one side nd dividing y the leding coefficient. (c) Explin why the fctors of this qudrtic re x+1 nd x p. Multiply these fctors together nd deduce equtions (9) nd (10). (4) Use Diophntos solution (10) to generte lots of pythgoren triples. Investigte Plimpton 322, nd find wht pythgoren triples re elieved to e written on it. Wht vlues of m yield these vlues?

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